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Florentin Smarandache, Surapati Pramanik
To cite this version:
Florentin Smarandache, Surapati Pramanik. New Trends in Neutrosophic Theory and Applications.
2016. �hal-01408066�
Pons Editions
(Editors)
New Trends in Neutrosophic Theory and Applications
Neutrosophic Science International Association President: Florentin Smarandache
Pons asbl
5, Quai du Batelage, Brussells, Belgium, European Union President: Georgiana Antonescu
DTP: George Lukacs
ISBN 978-1-59973-498-9
Florentin Smarandache, Surapati Pramanik (Editors)
New Trends
in Neutrosophic Theory and Applications
Pons Editions Brussels, Belgium, EU
2016
Pratt School of Engineering, Duke University, Durham, NC 27708, USA.
Dr. Mumtaz Ali, University of Southern Queensland, Australia.
Prof. Luige Vladareanu, Romanian Academy, Bucharest, Romania.
Dr. Dominique Maltese, SAFRAN ELECTRONICS & DEFENSE, Paris, France.
5
TABLE OF CONTENTS
Aims and Scope ... 7 Preface ... 9
DATA MINING
Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri
Role of Neutrosophic Logic in Data Mining ... 15
DECISION MAKING
Pranab Biswas, Surapati Pramanik, Bibhas C. Giri
Some Distance Measures of Single Valued Neutrosophic Hesitant Fuzzy Sets and Their Applications to Multiple Attribute Decision Making ... 27 Rıdvan Şahin, Peide Liu
Distance and Similarity Measures for Multiple Attribute Decision Making with Single-Valued Neutrosophic Hesitant Fuzzy Information ... 35 Pranab Biswas, Surapati Pramanik, Bibhas C. Giri
GRA Method of Multiple Attribute Decision Making with Single Valued Neutrosophic Hesitant Fuzzy Set Information ... 55 Partha Pratim Dey, Surapati Pramanik, Bibhas C. Giri
TOPSIS for Solving Multi-Attribute Decision Making Problems under Bi-Polar Neutrosophic Environment ... 65 Surapati Pramanik, Durga Banerjee, B. C. Giri
TOPSIS Approach for Multi Attribute Group Decision Making in Neutrosophic Environment ... 79 Kalyan Mondal, Surapati Pramanik, Florentin Smarandache
Several Trigonometric Hamming Similarity Measures of Rough Neutrosophic Sets and their Applications in Decision Making ... 93 Nital P. Nirmal, Mangal G. Bhatt
Selection of Automated Guided Vehicle using Single Valued Neutrosophic Entropy Based Novel Multi Attribute Decision Making Technique ... 105 Florentin Smarandache, Mirela Teodorescu
From Linked Data Fuzzy to Neutrosophic Data Set Decision Making in Games vs. Real Life ... 115 Partha Pratim Dey, Surapati Pramanik, Bibhas C. Giri
Extended Projection Based Models for Solving Multiple Attribute Decision Making Problems with Interval Valued Neutrosophic Information ... 127 Kanika Mandal, Kajla Basu
Multi Criteria Decision Making Method in Neutrosophic Environment Using a New Aggregation Operator, Score and Certainty Function ... 141 Surapati Pramanik, Shyamal Dalapati, Tapan Kumar Roy
Logistics Center Location Selection Approach Based on Neutrosophic Multi-Criteria Decision Making ... 161
E-LEARNING Nouran M. Radwan
Neutrosophic Applications in E-learning: Outcomes, Challenges and Trends ... 177
GRAPH THEORY
Said Broumi, Mohamed Talea,Assia Bakali, Florentin Smarandache
Single Valued Neutrosophic Graphs ... 187
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Said Broumi, Mohamed Talea,Assia Bakali, Florentin Smarandache
On Bipolar Single Valued Neutrosophic Graphs ... 203 Shimaa Fathi, Hewayda ElGhawalby, A.A. Salama
A Neutrosophic Graph Similarity Measures ... 223 Broumi, S., Smarandache, F., Talea, M., Bakali, A.
Operations on Interval Valued Neutrosophic Graphs ... 231
MEDICAL DIAGNOSIS
Deepika Koundal, Savita Gupta, Sukhwinder Singh
Applications of Neutrosophic Sets in Medical Image Denoising and Segmentation ... 257
NEUTROSOPHIC MODEL IN SOCIOLOGY Santanu Ku. Patro
On a model of Love dynamics: A Neutrosophic analysis ... 279
PROBABILITY THEORY A. A. Salama, Florentin Smarandache
Neutrosophic Crisp Probability Theory & Decision Making Process ... 291
TOPOLOGY
Francisco Gallego Lupiañez
On neutrosophic sets and topology ... 305 A.A. Salama, I.M.Hanafy, Hewayda Elghawalby, M.S.Dabash
Some GIS Topological Concepts via Neutrosophic Crisp Set Theory ... 315
OTHER THEORETICAL PAPERS J. Martina Jency, I. Arockiarani
Hausdorff Extensions in Single Valued Neutrosophic S∗ Centered Systems ... 327 Saeid Jafari, I. Arockiarani, J. Martina Jency
The Alexandrov-Urysohn Compactness On Single Valued Neutrosophic S∗
Centered Systems ... 345 Eman.M.El-Nakeeb, H. ElGhawalby, A.A.Salama, S.A.El-Hafeez
Foundation for Neutrosophic Mathematical Morphology ... 363 M. K. El Gayyar
Smooth Neutrosophic Preuniform Spaces ... 381 Hewayda ElGhawalby, A. A. Salama
Ultra neutrosophic crisp sets and relations ... 395 A.A.Salama, I.M.Hanafy, Hewayda Elghawalby, M.S.Dabash
Neutrosophic Crisp Closed Region and Neutrosophic Crisp Continuous Functions ... 403 Florentin Smarandache, Huda E. Khalid, Ahmed K. Essa
A New Order Relation on the Set of Neutrosophic Truth Values ... 413 Fu Yuhua
Expanding Comparative Literature into Comparative Sciences Clusters with Neutrosophy ... 415 Contributors ... 421
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Aims and Scope
Neutrosophic theory and applications have been expanding in all directions at an astonishing rate especially after the introduction the journal entitled “Neutrosophic Sets and Systems”. New theories, techniques, algorithms have been rapidly developed. One of the most striking trends in the neutrosophic theory is the hybridization of neutrosophic set with other potential sets such as rough set, bipolar set, soft set, hesitant fuzzy set, etc. The different hybrid structure such as rough neutrosophic set, single valued neutrosophic rough set, bipolar neutrosophic set, single valued neutrosophic hesitant fuzzy set, etc. are proposed in the literature in a short period of time.
Neutrosophic set has been a very important tool in all various areas of data mining, decision making, e-learning, engineering, medicine, social science, and some more.
The Book “New Trends in Neutrosophic Theories and Applications” focuses on theories, methods, algorithms for decision making and also applications involving neutrosophic information. Some topics deal with data mining, decision making, e- learning, graph theory, medical diagnosis, probability theory, topology, and some more.
Florentin Smarandache, Surapati Pramanik
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Preface
Neutrosophic set has been derived from a new branch of philosophy, namely Neutrosophy.
Neutrosophic set is capable of dealing with uncertainty, indeterminacy and inconsistent information. Neutrosophic set approaches are suitable to modeling problems with uncertainty, indeterminacy and inconsistent information in which human knowledge is necessary, and human evaluation is needed.
Neutrosophic set theory was proposed in 1998 by Florentin Smarandache, who also developed the concept of single valued neutrosophic set, oriented towards real world scientific and engineering applications. Since then, the single valued neutrosophic set theory has been extensively studied in books and monographs introducing neutrosophic sets and its applications, by many authors around the world. Also, an international journal - Neutrosophic Sets and Systemsstarted its journey in 2013.
Single valued neutrosophic sets have found their way into several hybrid systems, such as neutrosophic soft set, rough neutrosophic set, neutrosophic bipolar set, neutrosophic expert set, rough bipolar neutrosophic set, neutrosophic hesitant fuzzy set, etc. Successful applications of single valued neutrosophic sets have been developed in multiple criteria and multiple attribute decision making.
The present book starts by proposing an approach for data mining with single valued neutrosophic information from large amounts of data and then progresses to topics in decision making in neutrosophic environment and neutrosophic hybrid environment, e-learning, graph theory, medical diagnosis, neutrosophic models in sociology, topology, and some more.
The book collects thirty original research and application papers from different perspectives covering different areas of neutrosophic studies, such data mining, decision making, e-learning, graph theory, medical diagnosis, probability theory, topology, and some theoretical papers. This book shows examples applications of neutrosophic set and neutrosophic hybrid set in multiple criteria and multiple attribute decision making, medical diagnosis, etc.
The first chapter presents the two essential pillars of data mining: similarity measures and machine learning in single valued neutrosophic environment. It shows that neutrosophic logic can perform an important role in data mining method. It defines single valued neutrosophic score function (SVNSF) to aggregate attribute values of each alternative. It also presents an approach of data mining with single valued neutrosophic information from large amounts of data, and furnishes a numerical example for the proposed approach.
The second, third, and fourth chapter deal with decision making in neutrosophic hesitant fuzzy information.
The second chapter presents a class of distance measures for single-valued neutrosophic hesitant fuzzy sets and discusses their properties with parameter changing. It also provides multi-attribute decision making (MADM), an illustrative example, and a comparison with other existing methods.
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The third chapter presents an axiomatic system of distance and similarity measures between single-valued neutrosophic hesitant fuzzy sets. It also proposes a class of distance and similarity measures based on three basic forms: the geometric distance model, the set-theoretic approach, and the matching functions. The distance measure between each alternative and the ideal alternative is used to establish a multiple attribute decision making method under single-valued neutrosophic hesitant fuzzy environment. It provides a numerical example of investment alternatives to show the effectiveness and the usefulness of the proposed approach.
The fourth chapter extends grey relational analysis (GRA) method for MADM by defining score value, accuracy value, certainty value, and normalized Hamming distance of single-valued neutrosophic hesitant fuzzy set (SVNHFS). It also defines the positive ideal solution (PIS) and the negative ideal solution (NIS) by score value and accuracy value. It proposes GRA method for multi-attribute decision making under single valued neutrosophic hesitant fuzzy set environment.
It also provides an illustrative example to demonstrate the validity and the effectiveness of the proposed method.
The fifth chapter exposes TOPSIS method for MADM problems with bipolar neutrosophic information. The Hamming and the Euclidean distance functions are defined in order to determine the distance between bipolar neutrosophic numbers. The bipolar neutrosophic relative positive ideal solution (BNRPIS) and bipolar neutrosophic relative negative ideal solution (BNRNIS) are also characterized. The applicability of the proposed method is verified and a comparison with other existing methods is provided.
The sixth chapter presents TOPSIS approach for multi-attribute decision making in refined neutrosophic environment. An illustrative numerical example of tablet selection is provided to show the applicability of the proposed TOPSIS approach.
The seventh chapter presents several new similarity measures based on trigonometric Hamming similarity operators of rough neutrosophic sets and their applications in decision making. Some properties of the proposed similarity measures are established. Also, a numerical example is given to illustrate the applicability of the proposed similarity measures in decision making.
The eighth chapter develops a fuzzy single valued neutrosophic set with entropy weight based MADM technique. Its feasibility for automated guided vehicle (AGV) selecting and ranking of material handling systems for a given industrial application is examined.
The ninth chapter makes a connection between decision making in game and real life.
The tenth chapter develops two new methods for solving multiple attribute decision making (MADM) problems with interval valued neutrosophic assessments. It also discusses an alternative method to solve MADM problems based on the combination of angle cosine and projection method.
Finally, an illustrative numerical example in Khadi institution is provided to verify the effectiveness of the proposed methods.
The eleventh chapter introduces improved weighted average geometric operator and define new score function. It establishes a multiple-attribute decision-making method based the proposed operator and newly defined score function.
The twelfth chapter presents modeling of logistics center location problem using the score and accuracy function, hybrid-score-accuracy function of SVNNs and linguistic variables under single-
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valued neutrosophic environment, where weight of the decision makers are completely unknown and the weight of criteria are incompletely known.
The thirteenth chapter reports about current trends to enhance e-learning process by using neutrosophic techniques to extract useful knowledge for selecting, evaluating, personalizing, and adapting the eLearning process.
The fourteenth chapter introduces certain types of single valued neutrosophic graphs, such as strong single valued neutrosophic graph, constant single valued neutrosophic graph and complete single valued neutrosophic graphs. It investigates some of their properties with proofs and examples.
The fifteenth chapter combines the concept of bipolar neutrosophic set and graph theory. It introduces the notions of bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, complete bipolar single valued neutrosophic graphs, and regular bipolar single valued neutrosophic graphs. It also investigates some of their related properties.
The sixteenth chapter expounds two ways of determining the neutrosophic distance between neutrosophic vertex graphs. It proposes two neutrosophic distances based on the Haussdorff distance, and a robust modified variant of the Haussdorff distance. These distances satisfy the metric distance measure axioms. Furthermore, a similarity measure between neutrosophic edge graphs is explained, based on a probabilistic variant of Haussdorff distance.
The seventeenth chapter describes operations on interval valued neutrosophic graphs. It presents operations of Cartesian product, composition, union and join on interval valued neutrosophic graphs. It investigates some of their properties with proofs and examples.
The eighteenth chapter conveys the usefulness of neutrosophic theory in medical imaging, e.g.
denoising and segmentation.
The nineteenth chapter delivers a theoretical framework of love dynamics in neutrosophic environment.
The twentieth chapter emphasizes the neutrosophic crisp probability theory and the decision making process by presenting some fundamental definitions and operations.
The 21st and 22nd chapters devote to study neutrosophic sets and neutrosophic topology.
Chapters from 23rd to 30th present theoretical improvements of neutrosophic set and its variants.
We hope that this book will offer a useful resource of ideas, techniques, methods, and approaches for additional researches on applications in different fields of neutrosophic sets and various neutrosophic hybrid sets.
We are grateful to our referees, whose valuable and highly appreciated reviews guided us in selecting the chapters in the book.
Florentin Smarandache, Surapati Pramanik
D
ATAM
INING15
KALYAN MONDAL1,SURAPATI PRAMANIK2,BIBHAS C.GIRI3
1Department of Mathematics, Jadavpur University, West Bengal, India. E-mail: kalyanmathematic@gmail.com
²Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, PO-Narayanpur, and District: North 24 Parganas, Pin Code: 743126, West Bengal, India. E-mail: sura_pati@yahoo.co.in
3Department of Mathematics, Jadavpur University, West Bengal, India. E-mail: bcgiri.jumath@gmail.com
Role of Neutrosophic Logic in Data Mining
Abstract
This paper presents a data mining process of single valued neutrosophic information. This approach gives a presentation of data analysis common to all applications. Data mining depends on two main elements, namely the concept of similarity and the machine learning framework. It describes a lot of real world applications for the domains namely mathematical, medical, educational, chemical, multimedia etc. There are two main types of indeterminacy in supervised learning: cognitive and statistical. Statistical indeterminacy deals with the random behavior of nature. All existing data mining techniques can handle the uncertainty that arises (or is assumed to arise) in the natural world from statistical variations or randomness. Cognitive uncertainty deals with human cognition. In real world problems for data mining, indeterminacy components may arise. Neutrosophic logic can handle this situation. In this paper, we have shown the role of single valued neutrosophic set logic in data mining. We also propose a data mining approach in single valued neutrosophic environment.
Keywords
Data mining, single valued neutrosophic set, single valued neutrosophic score value.
1. Introduction
Data mining [1] is actually assumed as “knowledge mining” from data. Data mining is an essential process where intelligent methods are applied to extract data patterns [2]. Data mining is a process that analyzes large amounts of data to find new and hidden information. In other words;
it is the process of analyzing data from different perspectives and summarizing it into some useful information. The following are the different data mining techniques [3]: association, classification, clustering, and sequential patterns. E.Hullermeier [4] proposed fuzzy methods in data mining.
This paper focuses on real-world applications of single valued neutrosophic set [5] for data mining. Data mining decomposes into two main elements: the notion of similarity and the single valued neutrosophic machine learning techniques that are applied in the described applications.
Indeed, similarity, or more generally comparison measures are used at all levels of the data mining
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and information retrieval tasks. At the lowest level, they are used for the matching between a query to a database and the elements it contains, for the extraction of relevant data. Then similarity and dissimilarity measures can be used in the process of cleaning and management of missing data to create a reasonable set of data. To generalize particular information contained in this reasonable set, dissimilarity measures are used in the case of inductive learning and similarity measures for case-based reasoning or clustering tasks. Eventually, similarities are used to interpret results of the learning process into an expressible form of knowledge through the definition of prototypes. Most of collective data for an investigation involves indeterminacy. Single valued neutrosophic set can handle his situation. So, there is an important role of single valued neutrosophic set in data mining.
This paper is arranged as follows. Section 2 presents some basic knowledge of single valued neutrosophic set. Section 3 considers the component of similarity, and machine learning techniques.
Section 4 describes a methodical approach of data mining under single valued neutrosophic environment. Section 5 presents a numerical example for data mining. Section 6 presents concluding remarks.
2. Neutrosophic Preliminaries
2.1 Definition on neutrosophic sets [6]
The concept of neutrosophic set is originated from neutrosophy [6], a new branch of philosophy.
Definition 1:[6] Let ξ be a space of points (objects) with generic element in ξ denoted by x.
Then a neutrosophic set α in ξ is characterized by a truth membership function Tα an indeterminacy membership function Iα and a falsity membership function Fα. The functions Tα and Fα are real standard or non-standard subsets of 0,1that is Tα:0,1; Iα:0,1; Fα: 0,1.
It should be noted that there is no restriction on the sum of T x , I x , F x i.e.
3
0
T xI x F x
Definition 2: [6] The complement of a single valued neutrosophic set α is denoted by c and is defined by
x T x
T c
1 ;I c x I x
1
x F x
F c
1
Definition 3: (Containment) [6] A single valued neutrosophic set α is contained in the other single valued neutrosophic set β, if and only if the following result holds.
inf ,
infTx Tx supT x supT x
≥inf ,
infIx Ix supI x supI x
inf ,
infFx Fx supF x supF x
for all x in ξ.
Definition 4: (Single-valued single valued neutrosophic set)[5] .
Let ξ be a universal space of points (objects) with a generic element of ξ denoted by x.
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A single-valued single valued neutrosophic set S is characterized by a true membership function
),
Ts(x an indeterminacy membership function Is(x),a falsity membership functionFs(x)withTs(x),
)
Is(x , Fs(x)[0, 1] for all x in . When is continuous a SNVS can be written as
xTsx Fsx Isx x x
S , , ,
and when is discrete a SVNSs S can be written as:
T x F x I x x x
S S , S , S ,
It should be noted that for a SVNS S,
x F x I x x TS sup S sup S ≤3,∀
≤sup 0
and for a single valued neutrosophic set, the following relation holds:
x F x I x x TS sup S sup S ≤3,∀
≤sup 0
Definition 5: The complement of a single valued neutrosophic set S is denoted by Sc and is defined by
x F x
T c S
S ; I c x IS x
S 1 ; F c x TS x
S
Definition 6: A SVNS Sα is contained in the other SVNS Sβ , denoted as Sα Sβ iff,TS x TS x ;
x I x
IS S ; FS x FS x , x.
Definition 7: Two single valued single valued neutrosophic sets Sα and Sβ are equal, i.e. Sα = Sβ , if and only if SαSβ and Sα Sβ
Definition 8: (Union) The union of two SVNSs Sα and Sβ is a SVNSS, written asSSS. Its truth membership, indeterminacy-membership and falsity membership functions are related to those of Sand Sby
T x T x
TS(x)max S , S ;
x I x I x
IS max S , S ;
x F x F x
FS min S , S for all x in ξ
Definition 9: (intersection) The intersection of two SVNSs, Sα and Sβ is a SVNSS, written as
S S
S . Its truth membership, indeterminacy-membership and falsity membership functions are related to those of Sα an Sβ as follows:
x minT x,T x ;
TS S S
x maxI x,I x ;
IS S S
x F x F x x
FS max S , S ,
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3. Data Mining [2]
In this section, we discuss the theoretical background common to the applications, considering successively the notion of similarity and machine learning techniques under single valued neutrosophic environment.
3.1 Similarity [2]
The notion of similarity or more generally of comparison measures, is central for all real-world applications: it aims at quantifying the extent to which two objects are similar, or dissimilar, one to another, providing a numerical value for this comparison. Similarities and dissimilarities between objects are generally evaluated from values of their attributes or variables characterizing these objects. Dissimilarities are classically defined from distances. Similarities and dissimilarities are often expressed from each other: the more similar two objects are, the less dissimilar they are, the smaller their distance. Weights can be associated with variables, according to the semantics of the application or the importance of the variables. It appears that some quantities are used in various environments, with different forms, based on the same principles. Most of the classic dissimilarity measures between two objects with continuous numerical attributes are the Euclidian distance, the Manhattan distance, and more generally Minkowski distances.
3.2 Neutrosophy Machine Learning [2]
The second part of the theoretical background common to all applications concerns the neutrosophy machine learning techniques that use the previous similarity measures. Machine learning is an important way to extract knowledge from sets of cases, especially in large scale databases. In this section, we consider only the neutrosophy machine learning methods (involving indeterminacy) that are used in the applications, leaving aside other techniques as for neutrosophy case-based reasoning or neutrosophy association rules. Three methods are successively considered:
neutrosophy decision trees, neutrosophy prototypes and neutrosophy clustering. The first two belong to the supervised learning framework, i.e. they consider that each data point is associated with a category. Single valued neutrosophic set clustering belongs to the unsupervised learning framework, i.e. no decomposition of the data set with indeterminacy into categories is available.
3.2.1. Single valued neutrosophic set Decision Trees [2]
Neutrosophy decision trees (NDT) particularly can be interesting for data mining and information retrieval because they enable the user to take into account indeterminacy descriptions of the cases, or heterogeneous values (symbolic, numerical, or neutrosophical) [5]. Moreover, they are appreciated for their interpretability, because they provide a linguistic description of the relations between descriptions of the cases and decision to make or class to assign. The rules obtained through NDT make it easier for the user to interact with the system or the expert to understand, confirm or amend his own knowledge. Another quality of NDT is their robustness, since a small variation of descriptions does not drastically change the decision or the class associated with a case, which guarantees a resistance to measurement errors and avoids sharp differences for close values of the descriptions.
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3.2.2. Single valued neutrosophic set Prototype Construction [2]
Neutrosophy prototypes are another approaches to the characterization of data categories: they provide descriptions or interpretable summarizations of data sets, so as to help a user to better apprehend their contents: a prototype is an element chosen to represent a group of data, to summarize it and underline its most characteristic features. It can be defined from a statistical point of view, for instance as the data mean or the median; more complex representatives can also be used as the most typical value [7] for instance. The prototype notion was also studied from a cognitive science point of view, and specific properties were pointed out in [8]: it was shown that a prototype underlines the common features of the category members, but also their distinctive features as opposed to other categories, underlining the specificity of the group. Furthermore, prototypes were related to the typicality notion, i.e. the fact that all data do not have the same status as regards the group: some members of the group are better examples, more representative or more characteristic than others. It was also shown that the typicality of a point depends both on its resemblance to other members of the group (internal resemblance), and on its dissimilarity to members of other groups (external dissimilarity). More precisely, the method consists of computing internal resemblance and external dissimilarity for each data point. Internal resemblance and external dissimilarity are respectively defined as the aggregation (mean or median) of the resemblance to the other members of the group, and as the aggregation of the dissimilarity to members of other groups, for a given choice of the resemblance and dissimilarity measures.
4. Single Valued Neutrosophic Logic in Data Mining
The tools that have been proposed in single valued neutrosophic set (SVNS) have the potential to support all of the steps that neutralized a process of knowledge discovery. SVNS can be used in the data selection and preparation phase for data modeling. For any data analysis associated with an experiment or investigation, it is observed that much information involve indeterminacy. Single valued neutrosophic set logic is capable of dealing with this situation. So, for the case of data mining single valued neutrosophic set logic has an important role.
Standard methods of data analysis can be extended in a rather generic way by means of an extension principle. For example, the functional relation between the data points and the decision making function can be extended to the case of single valued neutrosophic data, where the observations are described in terms of single valued neutrosophic sets. If single valued neutrosophic data is not used in the data preparation phase, they can still be employed in a later stage in order to analyze the original data.
Various techniques are widely used for data mining from gathering data within a domain of expertise. Delphi method [9] and BIRCH method [10] are very popular for data mining. Rekha and Swapna [2] studied the role of fuzzy logic in data mining. Literature review reflects that there is no single valued neutrosophic approach for data mining till now.
4.1. Neutrosophic data mining method
Generally, there are many attributes in decision making problems, where some of them are important and others may not be so important. So it is crucial to select the proper attributes for decision-making situation. Now, we shall propose a methodical approach for data mining with
20
single valued neutrosophic information to prepare a panel of attributes which are technically sound.
All steps of this proposed approach are given as follows.
Step 1: Problem field selection
Consider a multi-attribute decision making problem with m alternatives and n attributes (large numbers of data). Let A1, A2, ..., Am and C1, C2, ..., Cn denote the alternatives and attributes respectively. In decision making process, we have to select a finite but more important attributes from given n attributes. All attributes are expressed in single valued neutrosophic number.
Table 1: Single valued neutrosophic set decision matrix
dij mn D
d d
A d
d d
A d
d d
A d
C C
C
mn m
m m
n n n
...
...
...
...
...
.
...
...
...
...
.
...
...
2 1
2 22
21 2
1 12
11 1
2
1
(1)
Here, dij(i = 1, 2, …, m and j = 1, 2, …, n) are all single valued neutrosophic numbers.
Step 2: Single valued neutrosophic set score matrix
Definition 10: Single valued neutrosophic score function (SVNSF)
Single valued neutrosophic score function (SVNSF) corresponding to each attribute is defined as follows.
mr rj rj rj
j T I F
C m
SVNSF 1
3 2 ) 1
( (2) Where, j = 1, 2, …, n
Using equation (2) we calculate single valued neutrosophic score matrix as follows.
Table: Single valued neutrosophic score matrix
) (Cj SVNSF
) (
) (
) (
2 2
1 1
n
n SVNSF C
C
C SVNSF C
C SVNSF C
value score ic neutrosoph valued
Single attributes
(3) Step 3: Selection zone
Single valued neutrosophic score values are classified into three zones. These are described as follows.
Definition 11: SVNSF of all the attributes are classified in three categories and it is defined as follows
Highly acceptable zone: 0.50 SVNSF(Cj) 1
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Tolerable acceptable zone: 0.25 SVNSF(Cj) 0.50 Unacceptable acceptable zone: 0.00 SVNSF(Cj) 0.25 Step 4: Ranking of attributes
According to the single valued neutrosophic score values, we can set up a panel of all attributes in descending order and we can choose important attributes from large number of attributes into decision making process considering highly acceptable zone and tolerable acceptable zone
Step 5: End
5. Numerical Example
In this section we demonstrate a numerical problem for applicability and effectiveness of this proposed approach. The methodical steps are as follows.
Step 1: Problem field selection
Suppose a person who wants to purchase a SIM card for mobile connection. So, it is necessary to select suitable SIM card for his/her mobile connection. There is a panel with four possible alternatives (SIM cards) for mobile connection. The alternatives (SIM cards) are presented as follows:
A1: Airtel A2: Vodafone A3: BSNL A4: IDEA.
For this purpose, the following attributes about SIM cards may be arise in decision making process. These are stated as follows.
1. Service quality of the corresponding company (C1) 2. Cost (C2)
3. Call rate per second (C3) 4. Internet facilities (C4) 5. Tower facility (C5) 6. Call drops (C6) 7. Risk factor (C7)
Table3: Single valued neutrosophic decision matrix
dij47 D
4 . 0 , 5 . 0 , 1 . 0 3 . 0 , 3 . 0 , 2 . 0 3 . 0 , 3 . 0 , 3 . 0 2 . 0 , 3 . 0 , 8 . 0 4 . 0 , 1 . 0 , 7 . 0 2 . 0 , 0 . 0 , 7 . 0 2 . 0 , 1 . 0 , 8 . 0
3 . 0 , 6 . 0 , 1 . 0 5 . 0 , 1 . 0 , 2 . 0 6 . 0 , 5 . 0 , 3 . 0 2 . 0 , 2 . 0 , 8 . 0 4 . 0 , 4 . 0 , 7 . 0 1 . 0 , 3 . 0 , 7 . 0 2 . 0 , 2 . 0 , 8 . 0
5 . 0 , 4 . 0 , 1 . 0 6 . 0 , 5 . 0 , 2 . 0 5 . 0 , 4 . 0 , 3 . 0 1 . 0 , 1 . 0 , 8 . 0 4 . 0 , 3 . 0 , 7 . 0 2 . 0 , 1 . 0 , 7 . 0 3 . 0 , 3 . 0 , 8 . 0
5 . 0 , 6 . 0 , 1 . 0 6 . 0 , 3 . 0 , 2 . 0 5 . 0 , 5 . 0 , 3 . 0 2 . 0 , 1 . 0 , 8 . 0 4 . 0 , 2 . 0 , 7 . 0 2 . 0 , 3 . 0 , 7 . 0 2 . 0 , 3 . 0 , 8 . 0
4 3 2 1
7 6
5 4
3 2
1
A A A A
C C
C C
C C
C
(4)
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Step 2: Single valued neutrosophic score matrix
Using equation (2) we calculate single valued neutrosophic score matrix as follows.
Table 4: Single valued neutrosophic set score matrix
) (Cj SVNSF
3833 . 0
4667 . 0
4667 . 0
8167 . 0
6833 . 0
7833 . 0
7833 . 0
7 6 5 4 3 2 1
C C C C C C C
value score ic neutrosoph valued
Single attributes
(5)
Step 3: Selection zone
Single valued neutrosophic score values are classified into three zones. These are described as follows.
Definition 11: SVNSF of all the attributes are classified in three categories and it is defined as follows
Highly acceptable zone: 0.50 SVNSF(Cj) 1 Tolerable acceptable zone: 0.25 SVNSF(Cj) 0.50 Unacceptable acceptable zone: 0.00 SVNSF(Cj) 0.25 Step 4: Ranking of attributes
From equation (5) we can write single valued neutrosophic score values of all attributes in descending order as follows.
) (C4
SVNSF SVNSF(C1) SVNSF(C2) SVNSF(C3)SVNSF(C5) SVNSF(C6) SVNSF(C7)
So, attributes corresponding to single values neutrosophic score values (highly acceptable and tolerance zone) can be chosen as important attributes for decision making process.
Step 5: End
6. Conclusion
In this paper we briefly present first two of the essential pillars of data mining: similarity measures and machine learning in single valued neutrosophic environment. We showed that neutrosophic logic can perform an important role in data mining method. We define single valued neutrosophic score function (SVNSF) to aggregate attribute values of each alternative. We also propose an approach for data mining with single valued neutrosophic information from large amounts of data and furnish a numerical example for the proposed approach. In future this method can be extended in interval neutrosophic environment for data mining.
23
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